He took over the job after Allardyce was caught up in a newspaper sting, in which he was alleged to have offered advice on how to "get around" rules on third-party player ownership. He was also criticized for some displays during the European Championship, which England exited embarrassingly at the last-16 stage at the hands of Iceland.
"I don't want to be a team that grinds things out", Southgate insisted.
"They had some chances, but we dominated the game", Hart said.
"I've been very impressed with him", said Southgate.
Hart was singled out for praise after he helped spare England embarrassment with a succession of outstanding saves, including a world-class reflex save when he turned Jasmin Kurtic's header on to the crossbar seconds after half-time. The chances that they had were set-plays or mistakes, so that's something that needs to be better.
And addressing the booing of Rooney at Wembley, Southgate said: "If you look at the number of games and his goal-scoring feats for England..."
Liverpool captain Jordan Henderson will lead the side out in Ljubljana as England look to maintain their ideal start to qualifying, with Tottenham Hotspur's Eric Dier - a specialist defensive midfielder - starting instead of Rooney.
Hart, who was shipped out on loan to Torino in Italy after being overlooked by new Manchester City boss Pep Guardiola, showed great agility to thwart the home side.
"We would have liked six points from these last two games but we have four and haven't conceded a goal".
Fortunately, the 29-year-old was able to continue and received plenty of plaudits for the stop. "But it's a very important point in the context of qualifying from the group".
Southgate added he is expecting a tough game from Slovenia, but is targeting an away victory that would keep England at the top of Group F and extend their 100 per cent record.
"We owe our goalkeeper really for getting a point", said Southgate. "We fought hard to get this point and maybe we deserved even more".